The Defocusing Nonlinear Schrodinger Equation with a Nonzero Background: Painleve Asymptotics in Two Transition Regions

In this paper, we address the Painleve asymptotics of the solution in two transition regions for the defocusing nonlinear Schrodinger (NLS) equation with finite density initial data iq(t) + q(xx) - 2(vertical bar q vertical bar(2) - 1)q = 0, q(x, 0) = q(0)(x) similar to +/- 1, x -> +/-infinity. The key to prove this result is the formulation and analysis of a Riemann-Hilbert problem associated with the Cauchy problem for the defocusing NLS equation. With the partial derivative-generalization of the Deift-Zhou nonlinear steepest descent method and double scaling limit technique, in two transition regions defined by P-+/- 1 := {(x, t) is an element of R x R+ : 0 < vertical bar x/2t - (+/- 1)vertical bar t(2/3) <= C}, where C > 0 is a constant, we find that the leading order approximation to the solution of the defocusing NLS equation can be expressed in terms of the Hastings-McLeod solution of the Painleve II equation in the generic case, while Ablowitz-Segur solution in the non-generic case.